Write the standard form of the equation of the circle described below. Center $(0,-1)$ passes through the point $(-1,-2)$

Step 1 ：The standard form of the equation of a circle is given by \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2 ：We know the center of the circle is \((0, -1)\), so \(h = 0\) and \(k = -1\).

Step 3 ：We can find the radius by calculating the distance between the center of the circle and the point it passes through, which is \((-1, -2)\).

Step 4 ：Using the distance formula, we find that \(r = \sqrt{2}\).

Step 5 ：Now that we have the radius, we can substitute the values of \(h\), \(k\), and \(r\) into the standard form of the equation of a circle to get the equation of the given circle.

Step 6 ：Substituting the values, we get \((x-0)^2 + (y+1)^2 = 2\).

Step 7 ：Final Answer: The standard form of the equation of the circle is \(\boxed{(x-0)^2 + (y+1)^2 = 2}\)